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- W4299971508 abstract "We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2pi$, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic cone-manifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group of the group of orientation-preserving isometries of the hyperbolic plane, and Markoff moves arising from the action of the mapping class group on the character variety." @default.
- W4299971508 created "2022-10-03" @default.
- W4299971508 creator A5007864871 @default.
- W4299971508 date "2010-06-27" @default.
- W4299971508 modified "2023-09-30" @default.
- W4299971508 title "Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori" @default.
- W4299971508 doi "https://doi.org/10.48550/arxiv.1006.5223" @default.
- W4299971508 hasPublicationYear "2010" @default.
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